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Fegen und Dünnheit mit Anwendungen auf die Laplace-und Wärmeleitungsgleichung

Wolfhard Hansen — 1971

Annales de l'institut Fourier

Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that E = R m + 1 is given the presheaf of solutions of i = 1 m u x i = u x m + 1 and B is a subset of R m × [ - , 0 ] satisfying { ( α x , α 2 t ) : ( x , t ) B , x R m , t R } B for α > 0 arbitrarily small. Then B is thin at 0 if and only if B is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several...

Abbildungen harmonischer Raüme mit Anwendung auf die Laplace und Wärmeleitungsgleichung

Wolfhard Hansen — 1971

Annales de l'institut Fourier

This paper is devoted to a study of harmonic mappings φ of a harmonic space E ˜ on a harmonic space E which are related to a family of harmonic mappings of E ˜ into E ˜ . In this way balayage in E may be reduced to balayage in E . In particular, a subset A of E is polar if and only if φ - 1 ( A ) is polar. Similar result for thinness. These considerations are applied to the heat equation and the Laplace equation.

Time-dependent Schrödinger perturbations of transition densities

Krzysztof BogdanWolfhard HansenTomasz Jakubowski — 2008

Studia Mathematica

We construct the fundamental solution of t - Δ y - q ( t , y ) for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We...

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